Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T14:05:53.083Z Has data issue: false hasContentIssue false
  • English
  • Français

THE AVERAGE DISTANCE BETWEEN TWO POINTS

Part of: General convexity

Published online by Cambridge University Press:  02 October 2009

BERNHARD BURGSTALLER  and
FRIEDRICH PILLICHSHAMMER
BERNHARD BURGSTALLER
Affiliation:
Doppler Institute for Mathematical Physics, Trojanova 13, 12000 Prague, Czech Republic (email: bernhardburgstaller@yahoo.de)
FRIEDRICH PILLICHSHAMMER*
Affiliation:
Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-4040 Linz, Austria (email: friedrich.pillichshammer@jku.at)
*
✉For correspondence; e-mail: friedrich.pillichshammer@jku.at
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide bounds on the average distance between two points uniformly and independently chosen from a compact convex subset of the s-dimensional Euclidean space.

Keywords

distance geometryrandom convex setsaverage distance

MSC classification

Secondary: 52A22: Random convex sets and integral geometry
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 353 - 359
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Alagar, V. S., ‘The distribution of the distance between random points’, J. Appl. Probab. 13 (1976), 558–566.CrossRefGoogle Scholar
[2]Anderssen, R. S., Brent, R. P., Daley, D. J. and Moran, P. A. P., ‘Concerning ∫ 10⋯∫ 10(x 21+⋯+x 2k)1/2 dx 1⋯ dx k and a Taylor series method’, SIAM J. Appl. Math. 30 (1976), 22–30.CrossRefGoogle Scholar
[3]Blaschke, W., ‘Eine isoperimetrische Eigenschaft des Kreises’, Math. Z. 1 (1918), 52–57.Google Scholar
[4]Dunbar, S. R., ‘The average distance between points in geometric figures’, College Math. J. 28 (1997), 187–197.CrossRefGoogle Scholar
[5]Gates, D. J., ‘Asymptotics of two integrals from optimization theory and geometric probability’, Adv. Appl. Probab. 17 (1985), 908–910.CrossRefGoogle Scholar
[6]Nickolas, P. and Yost, D., ‘The average distance property for subsets of Euclidean space’, Arch. Math. (Basel) 50 (1988), 380–384.CrossRefGoogle Scholar
[7]Pillichshammer, F., ‘A note on the sum of distances under a diameter constraint’, Arch. Math. (Basel) 77 (2001), 195–199.CrossRefGoogle Scholar
[8]Santaló, L. A., Integral Geometry and Geometric Probability, Encyclopedia of Mathematics and its Applications, 1 (Addison-Wesley, Reading, MA, 1976).Google Scholar
[9]Solomon, H., Geometric Probability, CBMS-NSF Regional Conference Series in Applied Mathematics, 28 (SIAM, Philadelphia, PA, 1978).CrossRefGoogle Scholar
[10]Wolf, R., ‘On the average distance property and certain energy integrals’, Ark. Mat. 35 (1997), 387–400.CrossRefGoogle Scholar